In turn the Jordan and Lie algebra give rise to a C*
algebra and we obtain quantum mechanics in the algebraic formalism. The
standard Hilbert space formulation is recovered by the GNS theorem/construction.

In the classical case (x=0) there are no Jordan
algebras, and in this case one has the regular function multiplication and the
Poisson bracket as realizations of the products θ and ρ.

What can we say about the third case, the hyperbolic
composability x=+1?

In this case we are lead to a hypothetical quantum mechanics
over split complex numbers.The
interesting part is that in this number system, the functional analysis is completely
changed because the norm triangle inequality which is the foundation of most of
the results in functional analysis is replaced by a reversed triangle
inequality (http://arxiv.org/pdf/1311.6461v2.pdf
). The key difference however between complex quantum mechanics (parabolic
composability) and split-complex quantum mechanics (hyperbolic composability)
is the lack of positivity. In
other words, we are not guaranteed to have positive probability predictions,
and we cannot define probabilities!!!
Hyperbolic composability violates one of the principles of nature introduced in
prior posts: positivity. Mathematically hyperbolic quantum mechanics is just as
rich and interesting as ordinary quantum mechanics, but it cannot correspond to
anything in nature. Only parabolic composability (classical mechanics) and
elliptic composability (quantum mechanics) can describe nature.

But how can we tell classical and quantum mechanics apart?
Simple: by experimental evidence in the form of violations of Bell inequalities. In classical mechanics, x=0 which means that the ontology always factorizes
neatly into system A and system B, but because x=-1 in quantum mechanics, this
factorization is no longer possible, and this is known as entanglement due to the superposition of the wavefunction. It is
superposition which allows for higher correlations than what one can expect
from any local realistic model.